Solve for $X$. $X+\left[\begin{array}{rr}-10 & 3 & -14 \\ 21 & 8 & -5 \\2 &6 &-1\end{array}\right]=\left[\begin{array}{rr}10 & 2 & 3 \\ 4 & -10 & 2 \\3 &4 &-1\end{array}\right] $ $X=$
The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $X+\left[\begin{array}{rr}-10 & 3 & -14 \\ 21 & 8 & -5 \\2 &6 &-1\end{array}\right]=\left[\begin{array}{rr}10 & 2 & 3 \\ 4 & -10 & 2 \\3 &4 &-1\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}-10 & 3 & -14 \\ 21 & 8 & -5 \\2 &6 &-1\end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}10 & 2 & 3 \\ 4 & -10 & 2 \\3 &4 &-1\end{array}\right] $ Then we can rewrite the equation as follows. $X+A=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}X+A&=B\\\\ X&=B-A \end{aligned}$ Finding $X$ We found that $X=B-A$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=B-A \\\\&=\left[\begin{array}{rr}10 & 2 & 3 \\ 4 & -10 & 2 \\3 &4 &-1\end{array}\right] -\left[\begin{array}{rr}-10 & 3 & -14 \\ 21 & 8 & -5 \\2 &6 &-1\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(10+10) & (2-3) & (3+14) \\ (4-21) & (-10-8) & (2+5) \\(3-2) &(4-6) &(-1+1)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}20 & -1 & 17 \\ -17 & -18 & 7 \\1 &-2 &0\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}20 & -1 & 17 \\ -17 & -18 & 7 \\1 &-2 &0\end{array}\right]$